Using Lemma REF  and Theorem REF  we obtain that \(\Phi _p(0), \Psi _{q}(0)\)  are c.n.u. contractions on the finite dimensional space \(\mathcal {E}\) , which further implies that \(\Phi (0),\Psi (0)\)  are pure contractions on \(\mathcal {E}\)  (see [1]}). Now, Theorem REF  implies that \(M_{\Phi _p}, M_{\Psi _q}\)  are pure isometries. This completes the proof.
